With obtained analytical expressions, in section ,
we analyze how the quadrupole contribution can be isolated by changing the inclination angle.
In the Conclusion, we discuss our final experimental proposal for the detection of the quadrupole (non-paraxial) contribution to the Smith-Purcell radiation. Our method is compatible both with moderate OAM values of \(\ell \sim 10\)  (see also [1]}, [2]}, [3]}) and large aspect ratio of Gaussian wave packets.
